Geodesics on the regular tetrahedron and the cube

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• 作者：Diana Davis^a ; ^{diana@math.northwestern.edu" class="auth_mail" title="E-mail the corresponding author} ; Victor Dods^b ; ^{vdods@ucsc.edu" class="auth_mail" title="E-mail the corresponding author} ; Cynthia Traub^c ; ^{ct@wustl.edu" class="auth_mail" title="E-mail the corresponding author} ; Jed Yang^d ; ^{jedyang@ucla.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Geodesic ; Cube ; Regular tetrahedron ; Stern&ndash ; Brocot tree
• 刊名：Discrete Mathematics
• 出版年：2017
• 出版时间：6 January 2017
• 年：2017
• 卷：340
• 期：1
• 页码：3183-3196
• 全文大小：599 K

文摘

Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern–Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than other pairs. We also obtain the fact that there are no geodesics that start and end at the same vertex on the regular tetrahedron or the cube.